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Dialogue: 0,0:00:00.26,0:00:04.71,Default,,0000,0000,0000,,What I want to do in this video is order these fractions from least to greatest
Dialogue: 0,0:00:04.71,0:00:10.38,Default,,0000,0000,0000,,And, the easiest way--and the way that people are sure to get the right answer--
Dialogue: 0,0:00:10.38,0:00:14.00,Default,,0000,0000,0000,,is to find a common denominator, because if we can't find a common denominator,
Dialogue: 0,0:00:14.00,0:00:21.43,Default,,0000,0000,0000,,these fractions are difficult to compare: 4/9 v. 3/4 v. 4/5, 11/12, 13/15.
Dialogue: 0,0:00:21.43,0:00:25.84,Default,,0000,0000,0000,,You can try to estimate them, but you'll be able to directly compare them if
Dialogue: 0,0:00:25.84,0:00:32.47,Default,,0000,0000,0000,,they all have the same denominator. So, the trick here is to first find the common denominator.
Dialogue: 0,0:00:32.47,0:00:36.43,Default,,0000,0000,0000,,And there is many ways to do it, you could just pick one of these numbers,
Dialogue: 0,0:00:36.43,0:00:42.05,Default,,0000,0000,0000,,and take all of its multiples until you find a multiple that is divisible by all of the rest.
Dialogue: 0,0:00:42.05,0:00:45.67,Default,,0000,0000,0000,,Another way to do it is to look at the prime factorization of each of these numbers.
Dialogue: 0,0:00:45.67,0:00:52.07,Default,,0000,0000,0000,,and then the 'least common multiple' of them would have each of those prime numbers in it.
Dialogue: 0,0:00:52.07,0:00:58.63,Default,,0000,0000,0000,,Let's do it that second way, and then verify it.
Dialogue: 0,0:00:58.63,0:01:08.43,Default,,0000,0000,0000,,So, 9 is 3{\i1}3, so our LCM is going to have at least one 3{\i0}3 in it.
Dialogue: 0,0:01:08.43,0:01:12.19,Default,,0000,0000,0000,,And then 4 is the same thing as 2*2.
Dialogue: 0,0:01:12.19,0:01:17.81,Default,,0000,0000,0000,,So, we will also have 2*2 in our prime factorization (LCM).
Dialogue: 0,0:01:17.81,0:01:22.36,Default,,0000,0000,0000,,5 is a prime number, so we'll put 5 right there.
Dialogue: 0,0:01:22.36,0:01:31.18,Default,,0000,0000,0000,,And then, 12 is the same thing as 2{\i1}6, and 6 = 2{\i0}3.
Dialogue: 0,0:01:31.18,0:01:40.87,Default,,0000,0000,0000,,So, in our LCM, we have to have two 2's, but we already have two 2's, and we already have one 3.
Dialogue: 0,0:01:40.87,0:01:48.18,Default,,0000,0000,0000,,Another way to think about it, is that something that is divisible by both 9 and 4
Dialogue: 0,0:01:48.18,0:01:50.20,Default,,0000,0000,0000,,is going to be divisible by 12.
Dialogue: 0,0:01:50.20,0:01:58.77,Default,,0000,0000,0000,,And then finally, we need it to be divisible by 15's prime factors.
Dialogue: 0,0:01:58.77,0:02:03.97,Default,,0000,0000,0000,,15 is the same thing as 3*5.
Dialogue: 0,0:02:03.97,0:02:09.31,Default,,0000,0000,0000,,So once again, we already have 3 and 5.
Dialogue: 0,0:02:09.31,0:02:15.16,Default,,0000,0000,0000,,So, this is our least common multiple (LCM).
Dialogue: 0,0:02:15.16,0:02:45.26,Default,,0000,0000,0000,,So, LCM is going to be equal to 3{\i1}3{\i0}2{\i1}2{\i0}5 =180
Dialogue: 0,0:02:45.26,0:02:52.87,Default,,0000,0000,0000,,So, our LCM is 180. So, we want to rewrite all of these fractions with 180 in the denominator.
Dialogue: 0,0:02:52.87,0:02:59.47,Default,,0000,0000,0000,,So, our first fraction, 4/9, is what over 180?
Dialogue: 0,0:02:59.47,0:03:04.06,Default,,0000,0000,0000,,To go from 9 to 180, we have to multiple 9 by 20.
Dialogue: 0,0:03:04.06,0:03:16.84,Default,,0000,0000,0000,,So, to get the denominator to equal 180, we multiple by 20.
Dialogue: 0,0:03:16.84,0:03:21.85,Default,,0000,0000,0000,,Since we don't want to change the value of the fraction, we should also multiple by the 4 by 20.
Dialogue: 0,0:03:21.85,0:03:28.86,Default,,0000,0000,0000,,4*20 = 80. So, 4/9 is the same thing as 80/180.
Dialogue: 0,0:03:28.86,0:03:37.20,Default,,0000,0000,0000,,Now, let's do 3/4. What do we have to multiple the denominator by to equal 180?
Dialogue: 0,0:03:37.20,0:03:42.66,Default,,0000,0000,0000,,You can divide 4 into 180 (180/4 = x) to figure that out.
Dialogue: 0,0:03:42.66,0:03:54.45,Default,,0000,0000,0000,,4*45 = 180. Now, you also have to multiple the numerator by 45.
Dialogue: 0,0:03:54.45,0:04:09.20,Default,,0000,0000,0000,,3*45 = 135. So, 3/4 equals 135/180.
Dialogue: 0,0:04:09.20,0:04:31.93,Default,,0000,0000,0000,,Now let's do 4/5. To get 180 from 5, multiple 5 by 36.
Dialogue: 0,0:04:31.93,0:04:35.13,Default,,0000,0000,0000,,Have to multiple numerator by same number, 36.
Dialogue: 0,0:04:35.13,0:04:46.32,Default,,0000,0000,0000,,So, 144/180.
Dialogue: 0,0:04:46.32,0:04:50.18,Default,,0000,0000,0000,,And then we have only two more to do.
Dialogue: 0,0:04:50.18,0:05:25.85,Default,,0000,0000,0000,,180/12 = 15. Same for numerator, 15. So, 11/12 = 165/180.
Dialogue: 0,0:05:25.85,0:05:28.07,Default,,0000,0000,0000,,And then finally, we have 13/15.
Dialogue: 0,0:05:28.07,0:05:51.43,Default,,0000,0000,0000,,To get 180 from 15, multiply 15 by 12--15{\i1}10 = 150, 30 remaining for 180. 15{\i0}2 = 30. So, 15*12 = 180.
Dialogue: 0,0:05:51.43,0:05:54.13,Default,,0000,0000,0000,,Multiple numerator by same number, 13.
Dialogue: 0,0:05:54.13,0:06:01.23,Default,,0000,0000,0000,,We know 12*12 = 144, so just add one more 12 = 156.
Dialogue: 0,0:06:01.23,0:06:08.43,Default,,0000,0000,0000,,So, we've rewritten each of these fractions with the new common denominator.
Dialogue: 0,0:06:08.43,0:06:13.03,Default,,0000,0000,0000,,Now, it is very easy to compare them. We only have to look at their numerators.
Dialogue: 0,0:06:13.03,0:06:21.43,Default,,0000,0000,0000,,Foe example, the smallest numerator is 80, so 4/9 is the least of these numbers.
Dialogue: 0,0:06:21.43,0:07:04.44,Default,,0000,0000,0000,,The next smallest number looks like 135, which was 3/4.
Dialogue: 0,0:07:04.44,0:07:08.52,Default,,0000,0000,0000,,And then the next one is going to be the 144/180, which was 4/5.
Dialogue: 0,0:07:08.52,0:07:20.83,Default,,0000,0000,0000,,Next is 156/180, which was 13/15.
Dialogue: 0,0:07:20.83,0:07:35.97,Default,,0000,0000,0000,,Finally, we have 165/180, which was 11/12.
Dialogue: 0,0:07:35.97,9:59:59.99,Default,,0000,0000,0000,,And, we're done! We have finished our ordering.